Research Initiation Award: Fermat-type Equations, their Resolutions, and Applications

The Historically Black Colleges and Universities Undergraduate Program (HBCU-UP) through Research Initiation Awards (RIA) provide support for junior and mid-career faculty at Historically Black Colleges and Universities who are building new research programs or redirecting and rebuilding existing research programs. It is expected that the award will help to further the faculty member's research capability and effectiveness and improve research and teaching at the home institution. This project investigates a class of long-standing mathematical problems known as Fermat-type equations, which seek whole-number and rational solutions to polynomial equations. These questions have played a central role in the development of modern number theory and continue to inspire new advances in mathematics. The research will develop new methods for studying families of equations that remain unsolved and will investigate how these methods can be applied to broader questions concerning the arithmetic of algebraic curves. Beyond advancing fundamental mathematical knowledge, the project will strengthen research capacity at Xavier University of Louisiana by engaging undergraduate students in research, fostering national and international collaborations, and disseminating results through publications, seminars, conference presentations, and a planned conference on Diophantine problems hosted at Xavier University of Louisiana. The project will develop and extend modern techniques for the study of generalized Fermat-type equations, with particular emphasis on families that remain inaccessible using existing methods. The research will combine theoretical and computational approaches to investigate the existence of integer and rational solutions, establish new nonexistence results, and develop methods applicable to related problems in arithmetic geometry. The project is expected to produce new mathematical techniques, computational implementations supporting the verification of theoretical results, and peer-reviewed publications. Computational resources developed during the project will be shared, where appropriate, to promote reproducibility and future research. Success will be measured through the development of new theoretical results, dissemination of research through publications and invited presentations, establishment of new collaborations, and the integration of undergraduate research and mentoring at Xavier University of Louisiana. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.